AITC01 22/3/04 14:45 Page 1 Chapter 1 THE BAYESIAN ALGORITHM An algorithm is a set of rules for doing a calculation. The Bayesian algorithm is a set of rules for using evidence (data) to change your beliefs. In this chapter we shall try to explain this algorithm. If this explanation is successful the reader may then put down the book and start doing Bayesian econometrics, for the rest of the book is little more than illustrative examples and technical details. Thus, chapter 1 and, to a lesser extent, chapter 2 are the most important parts of the book. We begin by explaining how we view an econometric analysis and by drawing a distinction between this and statistical analyses. 1.1 ECONOMETRIC ANALYSIS An econometric analysis is the confrontation of an economic model with evidence. An economic model usually asserts a relation between economic variables. For ex- ample, a model might assert that consumption, C, is linearly related to income, Y, so that C = α + βY for some pair of numbers α, β. Such a model is typically intended to provide a causal explanation of how some variables, for example C, are deter- mined by the values of others, for example Y. Typically, any model contains both potentially observable quantities, such as consumption and income, called (poten- tial) data; and it involves quantities, like α and β, that are not directly observable. Variables of this latter type are called parameters and will be denoted generically by the symbol θ. They are usually constrained to lie in a set to be denoted by Θ. In our example θ = (α, β) and the set Θ would normally be taken as two-dimensional euclidean space. Any value of θ, for example α = 10, β = 0.9, defines a particular structure, in this case C = 10 + 0.9Y, and the set of structures under consideration is said to be indexed by a parameter, θ. Evidence is provided by data on the operation of an economy. In the consumption/ income example relevant data would be provided by pairs of values for C and Y. There are usually many types of data that are relevant to any particular model. For example, we might have data on the consumption and income of different house- holds, or on the same household observed repeatedly, or on the aggregate income and consumption data of collections of households forming a region or a nation. AITC01 22/3/04 14:45 Page 2 2 The Bayesian Algorithm The objective of an econometric analysis is to answer two questions. The first question is whether the model is consistent with the evidence: this is called model criticism. This means asking whether any of the structures defined by the model are consistent with the evidence. In our example this would mean asking whether there is any parameter θ = (α, β), lying in Θ, such that, in our data, C = α + βY. The second question presumes that the answer to the first is “yes” and it asks what are the probabilities of the different structures defined by the model. Once this question has been answered the model can then be used for purposes of economic decision mak- ing, perhaps by a policy maker, perhaps by an individual economic agent. Such use will typically involve predicting the value of the variables for households or regions that are not included in the data. For example, given the structure θ = (10, 0.9) and told that Y = 100 then the economist would predict that C = 10 + 0.9 × 100 = 100. The practice of econometrics is, in fact, to ask these questions in reverse order. We begin by presuming that our model is consistent with the data and ask for the most likely structure in the light of the evidence. In traditional econometrics this θ ∈Θ involves forming a good estimate of 0 , the particular structure that is presumed to be, in some sense, true. In a Bayesian analysis this step involves using the data to form a probability distribution over the structures in Θ. An estimate, if one is required, might then be provided by reporting, for example, the most probable structure in the light of the evidence provided by the data. How then do we go about answering these questions in practice? In this chapter we shall focus on the second question in which we presume the consistency of the model with the data and ask how we determine the probabilities of the structures of which the model is composed. The method of doing this is to apply a theorem of probability, Bayes’ theorem, and here we shall describe in some detail how Bayes’ theorem is used to construct probabilities over alternative structures. In chapter 2 we shall describe some methods of answering the first question in which the investigator tries to decide whether the model is consistent with the evidence and if it is not, what to do next. 1.2 STATISTICAL ANALYSIS Statistical analysis deals with the study of numerical data. This is a largely descript- ive activity in which the primary aim is to find effective and economical repres- entations or summaries of such data. The point of the activity is to reduce the complexity of a set of numbers to a form which can be more easily comprehended.1 1 For instance, “. the object of statistical methods is the reduction of data. A quantity of data, which usually by its mere bulk is incapable of entering the mind, is to be replaced by relatively few quantities which shall adequately represent the whole, or which, in other words, shall contain as much as possible, ideally the whole, of the relevant information contained in the original data.” R. A. Fisher, On the mathematical foundations of theoretical statistics, Phil. Trans. Royal Soc., A222, 1922, p. 309, quoted in T. C. Koopmans, Linear Regression Analysis of Economic Time Series, Netherlands Economic Institute, Haarlem, 1937. AITC01 22/3/04 14:45 Page 3 The Bayesian Algorithm 3 The statistician summarizes data by calculating means, standard deviations, trends or regression lines; he represents data graphically by scatter diagrams, histograms, kernel smoothers and many other devices. He typically proposes and applies statis- tical models as simplified accounts of possible ways in which his data could have occurred. The application of such models involves estimating the parameters of such models and testing hypotheses about them. Statistical analysis is in many ways very close to econometrics, a subject which, to a statistician, can appear like a branch of applied statistics. Econometric technique is largely drawn from statistics and much of the content of this book will be famil- iar to a statistician. Indeed, in writing it I have drawn extensively on statistical books and articles. But there are profound differences between econometrics and statistics. The econometrician is primarily concerned with the analysis of the behavior of eco- nomic agents and their interactions in markets and the analysis of data is secondary to that concern. But markets can be in, or near, equilibrium; economic agents are presumed to be maximizing or minimizing some objective function; economic agents are often presumed to know relevant things that the econometrician does not. All these considerations tend to be fundamental to an econometric analysis and to dic- tate the class of models that are worth considering. They make the results of an eco- nometric analysis interpretable to the economist and give parameters solid meaning. Of course there is not and should not be a sharp line between econometrics and statistics; there is nothing at all wrong with an economist parsimoniously describing data or with a statistician trying to relate the parameters of his model to some under- lying theory. But the distinction between the disciplines exists, in my view, and should be kept in mind. 1.3 BAYES’ THEOREM Bayesian econometrics is the systematic use of a result from elementary probability, Bayes’ theorem. Indeed, from one angle, that’s all it is. There are not multiple methods of using numerical evidence to revise beliefs – there is only one – so this theorem is fundamental. What is Bayes’ theorem? When A and B are two events defined on a sample space the conditional probabil- ity that A occurs given that B has occurred is defined as PB( A) PAB()\ = (1.1) PB() as long as P(B) ≠ 0. Here P(B ʝ A) is the probability that both A and B occur and P(A\B) is the probability that A occurs given the knowledge that B has occurred. Equation (1.1) is true, of course, with A and B interchanged so that we also have P(B ʝ A) = P(B\A)P(A). Substituting this expression into (1.1) then gives AITC01 22/3/04 14:45 Page 4 4 The Bayesian Algorithm PBAPA()()\ PAB()\ = . (1.2) PB() When written in this form the definition is called Bayes’ theorem. It is a universally accepted mathematical proposition. But there is disagreement about its applicability to econometrics. Two related questions arise about Bayes’ theorem: one is about its interpretation and the other is about its use. Interpretations of probability The function P(.) has no interpretation in the mathematical theory of probability; all the theory does is define its properties. When probability theory is applied, as it is in econometrics, we need to decide how to interpret “the probability of an event.” In this book we shall take P(A) to measure the strength of belief in the proposition that A is true.2 Thus the larger P(A) the stronger your belief in the proposition that it represents.